A firm suffers a loss of Rs 144 if one of its special products does not sell. The original revenue is approximated by MR = 27-5x and marginal cost by MC = 4x - 27. Determine the profit function.

Answer

To determine the profit function based on the given marginal revenue (MR) and marginal cost (MC) functions, we can follow these steps:

### Step 1: Write down the given functions
The marginal revenue function is given by:
\[ MR = 27 - 5x \]
The marginal cost function is given by:
\[ MC = 4x - 27 \]

### Step 2: Find the marginal profit function
The marginal profit function (MP) can be found by subtracting the marginal cost from the marginal revenue:
\[ MP = MR - MC \]
Substituting the given functions:
\[ MP = (27 - 5x) - (4x - 27) \]
\[ MP = 27 - 5x - 4x + 27 \]
\[ MP = 54 - 9x \]

### Step 3: Integrate the marginal profit function
To find the profit function (P), we need to integrate the marginal profit function:
\[ P(x) = \int (54 - 9x) \, dx \]
Calculating the integral:
\[ P(x) = 54x - \frac{9}{2}x^2 + C \]
where \( C \) is the constant of integration.

### Step 4: Use the given information to find the constant
We know that the firm suffers a loss of Rs 144 when the product does not sell, which means when \( x = 0 \), the profit \( P(0) = -144 \):
\[ P(0) = 54(0) - \frac{9}{2}(0)^2 + C = -144 \]
This simplifies to:
\[ C = -144 \]

### Step 5: Write the final profit function
Now, substituting \( C \) back into the profit function:
\[ P(x) = 54x - \frac{9}{2}x^2 - 144 \]

Thus, the required profit function is:
\[ P(x) = 54x - \frac{9}{2}x^2 - 144 \]